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SC17-005: Implementation of the DPG Method in a FE Code Supporting H1, H(curl), H(div), and L2-Conforming Finite Elements

Leszek Demkowicz, The University of Texas at Austin
Stefan Henneking, The University of Texas at Austin

The four-lectures course is addressed to practitioners of standard Finite Element (FE) methods familiar with basic variational formulations, the (Bubnov–)Galerkin method and the standard technology of FEs. The class combines a short introduction to the “Discontinuous Petrov–Galerkin (DPG) Method with Optimal Test Functions” with a crash course on the energy spaces forming the exact sequence and the corresponding conforming FE discretizations. We will introduce the participants to hp3D—a 3D MPI/OpenMP code supporting hp-discretizations of the exact-sequence elements on hybrid (tets, cubes, prisms, pyramids) meshes and demonstrate how to implement the DPG method in such a framework. On the application side, we will focus on wave propagation problems: time-harmonic acoustics, Maxwell’s equations, and elastodynamics.


  1. Lecture 1
    1. Examples of variational formulations with symmetric and non-symmetric functional setting; brief introduction to energy spaces.
    2. A crash course on H1, H(curl), H(div), and L2-conforming finite elements.
  2. Lecture 2
    1. Introduction to the hp3D code.
    2. Examples of applications of the Bubnov–Galerkin method.
  3. Lecture 3
    1. A crash course on the DPG method.
    2. Implementation of DPG in the hp3D code.
  4. Lecture 4
    1. MPI/OpenMP parallel computation with the hp3D code.
    2. Examples of applications of the DPG method.

Course Material:

  1. L. Demkowicz. Lecture Notes on Energy Spaces. Technical Report 13, ICES, 2018.
  2. L. Demkowicz. Lecture Notes on Mathematical Theory of Finite Elements. Technical Report 11, Oden Institute, June 2020.
  3. S. Henneking and L. Demkowicz. Computing with hp Finite Elements III. Parallel hp Code. 2022. In preparation, available upon request.